Optimal periodic replenishment policies for spectrally positive L\'evy demand processes

We consider a version of the stochastic inventory control problem for a spectrally positive L'evy demand process, in which the inventory can only be replenished at independent exponential times. We show the optimality of a periodic barrier replenishment policy that restocks any shortage below a certain threshold at each replenishment opportunity. The optimal policies and value functions are concisely written in terms of the scale functions. Numerical results are also provided.